User:BobThompkins/Sandbox

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http://en.wikipedia.org/wiki/Help:Displaying_a_formula

$$X = \overline{B \cdot B}$$ $$X = \overline{B}$$ $$W = \overline{A \cdot A}$$ $$W = \overline{A}$$ $$Y = \overline{\overline{(A \cdot A)} \cdot \overline{(B \cdot B)}}$$

$$X = \overline{ \overline{(A \cdot B)} \cdot A}$$ $$Y = \overline{ \overline{(A \cdot B)} \cdot B}$$ $$Z = \overline{ \overline{ \overline{((A \cdot B)} \cdot A)} \cdot \overline{ \overline{((A \cdot B)} \cdot B)}}$$

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ELEC 2607 Winter 2012 Lab 2 3-bit binary sign-extended adder/subtractor

$$\begin{align} s_i & = \overline{y_i} \cdot \overline{x_i} \cdot c_i + \overline{y_i} \cdot x_i \cdot \overline{c_i} + y_i \cdot \overline{x_i} \cdot \overline{c_i} + y_i \cdot x_i \cdot c_i \\ & = \overline{y_i} (\overline{x_i} \cdot c_i + x_i \cdot \overline{c_i}) + y_i (\overline{x_i} \cdot \overline{c_i} + x_i \cdot c_i) \\ & = \overline{y_i} (x_i \oplus c_i) + y_i \overline{(x_i \oplus c_i)} \\ & = y_i \oplus x_i \oplus c_i \end{align}$$

$$\begin{align}

c_{i+1} & = \overline{y_i} \cdot x_i \cdot c_i + y_i \cdot \overline{x_i} \cdot c_i + y_i \cdot x_i \cdot \overline{c_i} + y_i \cdot x_i \cdot c_i \\ & = \overline{y_i} \cdot x_i \cdot c_i + y_i \cdot \overline{x_i} \cdot c_i + y_i \cdot x_i \cdot \overline{c_i} + y_i \cdot x_i \cdot c_i + y_i \cdot x_i \cdot c_i + y_i \cdot x_i \cdot c_i \\ & = x_i \cdot c_i (\overline{y_i} + y_i) + y_i \cdot c_i (\overline{x_i} + x_i) +  y_i \cdot x_i (\overline{c_i} + c_i) \\ & = x_i \cdot c_i + y_i \cdot c_i + y_i \cdot x_i \end{align}$$

$$Y - X = Y + \overline{X} + 1$$

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ELEC 2607 Winter 2012 Lab 4 MIDI Interface

$$MAJ = S_1S_2 + S_2S_3 + S_1S_3$$

$$\begin{align} GotStart & =  \overline{S_0}\overline{S_1}\overline{S_3} + \overline{S_0}\overline{S_1}\overline{S_2}+ \overline{S_0}\overline{S_2}\overline{S_3} \\ & = \overline{S_0}(\overline{S_1}\overline{S_3} + \overline{S_1}\overline{S_2} + \overline{S_2}\overline{S_3}) \\ & = \overline{S_0 + (S_1 + S_3)(S_1 + S_2)(S_2 + S_3)} \end{align}$$

$$\begin{align} D_2 & = Q_2 \oplus (Q_1Q_0) \\ D_1 & = Q_1 \oplus Q_0 \\ D_0 & = \overline{Q_0} \end{align}$$

$$\begin{align} D_3 & = \overline{Q_3}Q_2Q_1Q_0 \\ D_2 & = (\overline{Q_3}Q_1Q_0) \oplus (\overline{Q_3}Q_2) \\ D_1 & = (Q_1 \oplus Q_0) + \overline{Q_3} \\ D_0 & = \overline{Q_3}\overline{Q_0} \\ & = \overline{Q_3 + Q_0} \end{align}$$

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$$\oplus \bullet$$

$$Y = \overline{A \cdot B}$$ $$Y = \overline{B \cdot \overline{(A \cdot B)}}$$ $$Z = \overline{( \overline{(A \cdot \overline{(A \cdot B)})} \cdot \overline {(B \cdot \overline{(A \cdot B)})})}$$ $$X = \overline{\overline{(C \oplus (A \oplus B)} \cdot C) \cdot \overline{(B \cdot (A \oplus B)})}$$

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$$Y = A \oplus B$$ $$X = (A \oplus B) \cdot B$$ $$X = \overline {\overline{(A \oplus B) \cdot B} \cdot \overline {(C \cdot (C \oplus (A \oplus B)}}$$ $$Y = (A \oplus B) \oplus C$$

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$$\begin{align} T & = W \cdot \bar{S1} \cdot \bar{S0} + X \cdot \bar{S1} \cdot S0 + Y \cdot S1 \cdot \bar{S0} + Z \cdot S1 \cdot S0\\ & = (W \cdot \bar{S0} + X \cdot S0) \cdot \bar{S1} + (Y \cdot \bar{S0} + Z \cdot S0) \cdot S1\\ \end{align}$$

$$(D1)$$

$$\begin{align} A & = T \cdot \bar{P2} \cdot \bar{P1} \cdot \bar{P0} \\ & = \bar{T \cdot \bar{P2}} + P1 + P0 \\ B & = T \cdot \bar{P2} \cdot \bar{P1} \cdot P0 \\ & = \bar{T \cdot \bar{P2}} + P1 + \bar{P0} \\ C & = T \cdot \bar{P2} \cdot P1 \cdot \bar{P0}\\ & = \bar{T \cdot \bar{P2}} + \bar{P1} + P0 \\ D & = T \cdot \bar{P2} \cdot P1 \cdot P0 \\ & = \bar{T \cdot \bar{P2}} + \bar{P1} + \bar{P0} \\ E & = T \cdot P2 \cdot \bar{P1} \cdot \bar{P0} \\ F & = T \cdot P2 \cdot \bar{P1} \cdot P0 \\ G & = T \cdot P2 \cdot P1 \cdot \bar{P0} \\ H & = T \cdot P2 \cdot P1 \cdot P0 \\ \end{align}$$

$$(D1)$$